IJPAM: Volume 80, No. 3 (2012)

ON SOME CYCLIC HOMOGENEOUS POLYNOMIAL
INEQUALITIES OF DEGREE FOUR IN
REAL VARIABLES UNDER CONSTRAINTS

Vasile Cirtoaje$^1$, Vo Quoc Ba Can$^2$
$^1$Department of Automatic Control and Computers
University of Ploiesti
100680, Ploiesti, ROMANIA
$^2$Can Tho University of Medicine and Pharmacy
VIETNAM


Abstract. We find the best lower and upper bounds of the ratio

\begin{displaymath}\frac{x^3y+y^3z+z^3x}{(x^2+y^2+z^2)^2}\end{displaymath}

for all real numbers $x,y,z$ satisfying $k(x^2+y^2+z^2)=xy+yz+zx,$ or $k(x^2+y^2+z^2)\ge xy+yz+zx,$ or $k(x^2+y^2+z^2)\le xy+yz+zx,$ where $k$ is a given real number. The obtained results generalize the known cyclic inequalities

\begin{displaymath}(x^2+y^2+z^2)^2\ge 3(x^3y+y^3z+z^3x)\end{displaymath}

and

\begin{displaymath}(x^2+y^2+z^2)^2+\frac 8{\sqrt 7}(x^3y+y^3z+z^3x)\ge 0,\end{displaymath}

which hold for all real $x,y,z$.

Received: January 28, 2012

AMS Subject Classification: 26D05

Key Words and Phrases: cyclic homogeneous inequality, fourth degree polynomial, three real variables under constraints, equality conditions

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 80
Issue: 3